Having established the form of the optimal contract, the planner’s problem is equivalent to findingmandMthat minimize
M(1−F(M))+ Z M 0 v f(v)d v+ζ¯F(m) Z m 0 (m−v)f(v)d v, (37) subject to the concessionaire’s participation constraint (11). Noting that (11) implicitly defines M as a function ofm, we have that:
M0(m)= − F(m)u
0(m−I)
(1−F(M))u0(M−I). (38)
A similar calculation shows that the rate at whichMandmhave to change to keep the objective function (37) unchanged is given by
M0(m)= −
¯
ζF(m)
1−F(M). (39)
Equating (38) and (39) forM0(m) leads to (12) and completes the proof.58
C.3 Proof of Proposition 5
With the assumptions and notation introduced in the main text we prove that:
M0(ζ)= λF(m) (λζ−α)F(m)CARA(M−I)+(λ−α)(1−F(M))CARA(m−I), m0(ζ)= − λ(λ−α)(1−F(M)) (λζ−α)[(λζ−α)F(m)CARA(M−I)+(λ−α)(1−F(M))CARA(m−I)], M0(ζ)−m0(ζ)= λ[(λζ−α)F(m)+(λ−α)(1−F(M))] (λζ−α)[(λζ−α)F(m)CARA(M−I)+(λ−α)(1−F(M))CARA(m−I)].
It follows that risk borne by the concessionaire increases with the social cost of subsidies,ζ. Furthermore, (λζ−α)(M0(ζ)−m0(ζ))/λtakes a value between 1/CARA(m−I) and 1/CARA(M−I).
We defineC(I)≡CARA(M−I)/CARA(m−I) and also show that:
m0(I)=1+ ¯ ζC(I)RM m u0(v−I)f(v)d v [ ¯ζC(I)F(m)+1−F(M)]u0(m−I), M0(I)=1+ RM m u0(v−I)f(v)d v [ ¯ζC(I)F(m)+1−F(M)]u0(M−I), M0(I)−m0(I)=ζ¯(1−C(I)) RM m u0(v−I)f(v)d v [ ¯ζC(I)F(m)+1−F(M)]u0(m−I).
It follows thatmandM grow faster thanI. Also, for a concessionaire with decreasing absolute risk aversion, the wedge betweenM andmincreases withI, while it does not depend onI for a concessionaire with constant absolute risk aversion.
Proof Implicit differentiation of (12) with respect toζand a bit of algebra leads to: M0(ζ)= λ
(λζ−α)CARA(M−I)+
CARA(m−I) CARA(M−I)m
0(ζ).
Implicitly differentiating (11) with respect to ¯ζleads to: M0(ζ)= − (λζ−α)F(m)
(λ−α)(1−F(M)))m 0(ζ).
Both expressions above lead to the comparative statics results for ¯ζ. Implicit differentiation of (12) with respect toIleads to:
m0(I)−1
M0(I)−1=ζ¯C(I). Implicit differentiation of (11) with respect toI leads to:
F(m)u0(m−I)[m0(I)−1]+
Z M
m
u0(v−I)f(v)d v+(1−F(M))u0(M−I)[M0(I)−1]=0. The three comparative statics expressions inI now follow easily.
C.4 Proof of Proposition 9
Proof Part (i) It follows immediately from the planner’s objective function thatpG(θ)=p∗(λ,θ) whenγ<1, that is, when the contract length is finite.
To derive the expressions forpC(θ), consider first the case where the contract length is finite. We fix the concessionaire’s profits, and choose the price that maximizes the planner’s welfare, that is, we solve:
max
p,γ γH(p,α)+(1−γ)H ∗(λ) s.t. γΠ(p)=K,
where we have droppedθfrom our notation,H∗(λ)≡H(p∗(λ)) andpstands forpC. Using the constraint to get rid ofγin the objective function leads to the following equivalent problem:
max
p
CS(p)−H∗(λ)
The corresponding first order condition leads to: H∗(λ)=CS(p)−CS
0(p)
Π0(p)Π(p) and it follows from (16) thatp=p∗(λ) is optimal in this case.
Part (ii) Next we consider the case whereS>0 and maximize the planner’s objective function overpandS, keeping fixed the concessionaire’s total profits:
max
p,S H(p,α)−(λζ−α)S
s.t. Π(p)+S=K.
This time we use the constraint to get rid ofSin the objective function, which leads to: max
p H(p,α)+(λζ−α)Π(p),
which, by the definition ofH, is equivalent to choosing the user fee that maximizesH(p,λζ). It follows thatpC =p∗(λζ) in this case.
Part (iii) We consider two intermediate demand states,θ1andθ2, and find the optimal price
in each state subject to a fixed expected utility for the concessionaire. That is, we solve: max p1,p2 H(p1,α,θ1)f(θ1)+H(p2,α,θ2)f(θ2) s.t. u¡Π(p1,θ)−I ¢ f(θ1)+u ¡ Π(p2,θ)−I ¢ f(θ2)=K.
The Lagrangian for this problem is
L(p1,p2)=H(p1,α,θ1)f(θ1)+H(p2,α,θ2)f(θ2)+µ[u10f(θ1)+u02f(θ2)],
whereui0=u(Π(pi,v)−I),i=1, 2, andµdenotes the multiplier for the concessionaire’s partici-
pation constraint.
Using the first order conditions inp1andp2to get rid ofµthen leads to:
u10 u20 = CSp(p1,θ) Πp(p1,θ1)+α CSp(p2,θ2) Πp(p1,θ1) +α .
Defineη1andη2viap1=p∗(η1,θ1) andp2=p∗(η2,θ2). Sinceθ1andθ2are intermediate de-
expression combined with (16) implies that: u10 u20 = η1−α η2−α .
A similar argument, with an intermediate and a low (high) demand state instead of two inter- mediate states, leads to the second (third) equality in (20).
C.5 Proof of Proposition 10
We use Figure 2 to extend (37) and (38) to the more general setting considered here and in this way prove (22). We show that the planner substitutesmandM at a rate:
M0(m)= −
¯
ζFλζ(m)
1−Fλ(M), (40)
while the rate at whichm andM are substituted along the concessionaire’s participation con- straint satisfies:
M0(m)= − Fλζ(m)u
0(m−I)
(1−Fλ(M))u0(M−I). (41)
Equating both rates of substitution leads to (22).
Consider the impact on the concessionaire’s participation constraint of an increase ofmby
∆m. Demand states than originally enjoyed a minimum revenue guarantee ofmsee this guar- antee increase by∆m, thereby increasing the concessionaire’s expected utility byFλζ(m)u0(m−
I)∆m. We also have a small fraction of states—those withvλζ∈[m,m+∆m]—that now have a guarantee and did not have one before. And the user-fee in these states is somewhat smaller once they have a minimum revenue guarantee. In any case, the contribution of these marginal states to the concessionaire’s expected utility is of second order in∆m and can therefore be ignored.
A similar argument shows that a decrease ofM by∆M leads to a decrease of the conces- sionaire’s expected utility of (1−Fλ(M)u0(M−I)∆M, where again we ignore higher order terms in∆M. Equating to zero the expected utility change associated with an increase inm and a decrease ofM leads to (41).
To derive (40) we first use our two-threshold characterization of the optimal contract to sim- plify the planner’s objective function (17a). In high demand states we haveγΠ(p∗(λ))=Mand therefore
[αγ+λ(1−γ)]Π=λΠ−(λ−α)M.
We use this expression to get rid ofγin the expression for welfare in high demand states:
Whigh=CS(p∗(λ))+αM + λ(Π(p∗(λ))−M). (42)
function for these states:
Wlow=CS(p∗(λζ))+αm +λζ(Π(p∗(λζ))−m) (43)
Finally, in intermediate demand states we have:
Wint=CS(p∗(η))+αΠ(p∗(η)), (44)
withη∈(λ,λζ) determined from (20).
Consider next the effect on total welfare of an increase of∆m inm and a decrease of∆M inM. Comparing (42)–(44) it is clear that the change in welfare due to marginal firms—those close tomorM—is second order, sinceη≈λζfor firms withwλζclose tomandη≈λfor firms withwλclose toM. It follows that, as in the previous case, the first order aggregate change in welfare is due to inframarginal low demand states and inframarginal high demand states. The subsidy paid out in the former states increases significantly, leading to a welfare reduction of (λζ−α)Fλζ(m)∆m. And user fees freed up by the decrease inMallow the government to reduce distortions elsewhere in the economy, increasing welfare by (λ−α)(1−Fλ(M))∆M. Equating to zero the total change in welfare leads to (40) and completes the proof.